Existence of multiple solutions for a singular quasilinear elliptic equation in RNRN

In this paper, we are concerned with the existence of multiple positive solutions for the singular quasilinear elliptic problem in RNRN{−div(|x|−ap|∇u|p−2∇u)+g(x)|u|p−2u=h(x)|u|m−2u+λH(x)|u|n−2u,x∈RNu(x)>0,x∈RN. where λ>0λ>0 is a real parameter and 1010. The weight function g(x)g(x) is a bounded nonnegative function with ‖g‖∞>0‖g‖∞>0 and h(x),H(x)h(x),H(x) are continuous functions which change sign in RNRN. We prove that there admits at least two positive solutions by using the Nehari manifold and the fibrering maps associated with the Euler functional for this problem.